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## Limiting Cases

Let's consider a couple of "limiting cases" of such solutions. First, suppose that the linear restoring force is extremely weak compared to the "drag" force - i.e.13.7   . Then    and the solutions are   [i.e.   constant, plausible only if  x = 0] and   ,  which gives the same sort of damped behaviour as if there were no restoring force, which is what we expected.

Now consider the case where the linear restoring force is very strong and the "drag" force extremely weak - i.e.   . Then    and the solutions are   ,  or13.8

 x(t) = (13.23) (13.24) = (13.25)

where   . We may then think of    as a complex frequency13.9 whose real part is    and whose imaginary part is  . What sort of situation does this describe? It describes a weakly damped harmonic motion in which the usual sinusoidal pattern damps away within an "envelope" whose shape is that of an exponential decay. A typical case is shown in Fig. 13.5.

Next: Generalization of SHM Up: Damped Harmonic Motion Previous: Damped Harmonic Motion
Jess H. Brewer - Last modified: Sun Nov 15 13:49:43 PST 2015